Theorem tz7.48-2 2995

Description: Proposition 7.48(2) of [TakeutiZaring] p. 51.

Hypothesis

Ref	Expression
tz7.48.1	|- F Fn On

Assertion

Ref	Expression
tz7.48-2	|- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)

Distinct variable group(s):   x,F   x,A

Proof of Theorem tz7.48-2

Step	Hyp	Ref	Expression
1		dmres 2584	. . . . . . . . . . . . . . . 16 |- dom (F |` x) = (x i^i dom F)
2	1	eleq2i 1153	. . . . . . . . . . . . . . 15 |- (y e. dom (F |` x) <-> y e. (x i^i dom F))
3		elin 1635	. . . . . . . . . . . . . . 15 |- (y e. (x i^i dom F) <-> (y e. x /\ y e. dom F))
4	2, 3	bitr 151	. . . . . . . . . . . . . 14 |- (y e. dom (F |` x) <-> (y e. x /\ y e. dom F))
5		tz7.48.1	. . . . . . . . . . . . . . . . 17 |- F Fn On
6		fnfun 2721	. . . . . . . . . . . . . . . . 17 |- (F Fn On -> Fun F)
7	5, 6	ax-mp 6	. . . . . . . . . . . . . . . 16 |- Fun F
8		funres 2697	. . . . . . . . . . . . . . . 16 |- (Fun F -> Fun (F |` x))
9	7, 8	ax-mp 6	. . . . . . . . . . . . . . 15 |- Fun (F |` x)
10		fvrn 2888	. . . . . . . . . . . . . . 15 |- ((Fun (F |` x) /\ y e. dom (F |` x)) -> ((F |` x)` y) e. ran (F |` x))
11	9, 10	mpan 518	. . . . . . . . . . . . . 14 |- (y e. dom (F |` x) -> ((F |` x)` y) e. ran (F |` x))
12	4, 11	sylbir 176	. . . . . . . . . . . . 13 |- ((y e. x /\ y e. dom F) -> ((F |` x)` y) e. ran (F |` x))
13		fvres 2840	. . . . . . . . . . . . . . . 16 |- (y e. x -> ((F |` x)` y) = (F` y))
14	13	eleq1d 1155	. . . . . . . . . . . . . . 15 |- (y e. x -> (((F |` x)` y) e. ran (F |` x) <-> (F` y) e. ran (F |` x)))
15		df-ima 2431	. . . . . . . . . . . . . . . 16 |- (F"x) = ran (F |` x)
16	15	eleq2i 1153	. . . . . . . . . . . . . . 15 |- ((F` y) e. (F"x) <-> (F` y) e. ran (F |` x))
17	14, 16	syl6rbbr 417	. . . . . . . . . . . . . 14 |- (y e. x -> ((F` y) e. (F"x) <-> ((F |` x)` y) e. ran (F |` x)))
18	17	adantr 306	. . . . . . . . . . . . 13 |- ((y e. x /\ y e. dom F) -> ((F` y) e. (F"x) <-> ((F |` x)` y) e. ran (F |` x)))
19	12, 18	mpbird 171	. . . . . . . . . . . 12 |- ((y e. x /\ y e. dom F) -> (F` y) e. (F"x))
20		eleq1a 1158	. . . . . . . . . . . . 13 |- ((F` y) e. (F"x) -> ((F` x) = (F` y) -> (F` x) e. (F"x)))
21		eldifn 1592	. . . . . . . . . . . . 13 |- ((F` x) e. (A \ (F"x)) -> -. (F` x) e. (F"x))
22	20, 21	nsyli 106	. . . . . . . . . . . 12 |- ((F` y) e. (F"x) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
23	19, 22	syl 12	. . . . . . . . . . 11 |- ((y e. x /\ y e. dom F) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
24		fndm 2723	. . . . . . . . . . . . 13 |- (F Fn On -> dom F = On)
25	5, 24	ax-mp 6	. . . . . . . . . . . 12 |- dom F = On
26	25	eleq2i 1153	. . . . . . . . . . 11 |- (y e. dom F <-> y e. On)
27	23, 26	sylan2br 348	. . . . . . . . . 10 |- ((y e. x /\ y e. On) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
28		pm3.26 256	. . . . . . . . . 10 |- ((y e. x /\ x e. On) -> y e. x)
29		onelon 2223	. . . . . . . . . . 11 |- ((x e. On /\ y e. x) -> y e. On)
30	29	ancoms 334	. . . . . . . . . 10 |- ((y e. x /\ x e. On) -> y e. On)
31	27, 28, 30	sylanc 361	. . . . . . . . 9 |- ((y e. x /\ x e. On) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
32	31	exp 291	. . . . . . . 8 |- (y e. x -> (x e. On -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y))))
33	32	imp3a 279	. . . . . . 7 |- (y e. x -> ((x e. On /\ (F` x) e. (A \ (F"x))) -> -. (F` x) = (F` y)))
34	33	com12 13	. . . . . 6 |- ((x e. On /\ (F` x) e. (A \ (F"x))) -> (y e. x -> -. (F` x) = (F` y)))
35	34	r19.21aiv 1259	. . . . 5 |- ((x e. On /\ (F` x) e. (A \ (F"x))) -> A.y e. x -. (F` x) = (F` y))
36	35	exp 291	. . . 4 |- (x e. On -> ((F` x) e. (A \ (F"x)) -> A.y e. x -. (F` x) = (F` y)))
37	36	r19.20i 1253	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> A.x e. On A.y e. x -. (F` x) = (F` y))
38		ssid 1519	. . . 4 |- On (_ On
39	5	tz7.48lem 2993	. . . 4 |- ((On (_ On /\ A.x e. On A.y e. x -. (F` x) = (F` y)) -> Fun `'(F |` On))
40	38, 39	mpan 518	. . 3 |- (A.x e. On A.y e. x -. (F` x) = (F` y) -> Fun `'(F |` On))
41	37, 40	syl 12	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'(F |` On))
42		fnrel 2722	. . . . . 6 |- (F Fn On -> Rel F)
43	5, 42	ax-mp 6	. . . . 5 |- Rel F
44	25, 38	eqsstr 1530	. . . . 5 |- dom F (_ On
45		relssres 2596	. . . . 5 |- ((Rel F /\ dom F (_ On) -> (F |` On) = F)
46	43, 44, 45	mp2an 520	. . . 4 |- (F |` On) = F
47		cnveq 2513	. . . 4 |- ((F |` On) = F -> `'(F |` On) = `'F)
48	46, 47	ax-mp 6	. . 3 |- `'(F |` On) = `'F
49		funeq 2683	. . 3 |- (`'(F |` On) = `'F -> (Fun `'(F |` On) <-> Fun `'F))
50	48, 49	ax-mp 6	. 2 |- (Fun `'(F |` On) <-> Fun `'F)
51	41, 50	sylib 173	1 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   e. wel 803   = wceq 1091   e. wcel 1092  A.wral 1201   \ cdif 1484   i^i cin 1486   (_ wss 1487  Oncon0 2199  `'ccnv 2409  dom cdm 2410  ran crn 2411   |` cres 2412  "cima 2413  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  tz7.48-3 2996

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438

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